I come up with a confusion about understanding a part of a paper. I just write it here with simplification as follows:
Let $x_{i}$ for $i=1,\ldots,K$ be the independent realizations of $x$, a zero mean complex random variable with unit variance. Consider $ \hat{p} = \sum_{i=1}^{K} x_{i} x_{i}^{*}.$ Then $$ \mathbb{E}\{\hat{p}\hat{p}^{*}\} = K \beta, \quad \beta \triangleq \mathbb{E}\{|x|^4\}. $$
How does he obtain this? Does it correct to assume that $\mathbb{E}\{|x_{i}|^4\}$ is equal for all realizations?
"$x_i$ is a realization of $x$" means that $x_i$ and $x$ have the same distribution. So expected value of any function of $x_i$ must be the same as expected value of the same function of $x$. Note also that expected value of a sum is the sum of the expected values (i.e. expected value is linear).
I don't see how you get the $4$'th power, though. Is there a typo?